Lemma 2.3 ([1]}) 
The cyclotomic units of \(K_m\)\((\) resp. \(K_m^+\)\()\)  are generated by \(\zeta _{ 2^{m}}\)\((\) resp. \(-1\)\()\)  and \(\xi _{k,m}=\zeta _{2^m}^{(1-k)/2}\frac{1-\zeta _{2^m}^k}{ 1-\zeta _{2^m}},\)  where \(k\)  is an odd integer such that \(1< k< 2^{m-1}\) .

